关于带附加零元的半群 $\mathscr{C}_{+}(a,b)$ 和 $\mathscr{C}_{-}(a,b)$ 上的局部紧移位连续拓扑

Congruence topologies on the mapping class group

Let $Γ(S)$ be the pure mapping class group of a connected orientable surface $S$ of negative Euler characteristic. For ${\mathscr C}$ a class of finite groups, let $\hatπ_1(S)^{\mathscr C}$ be the pro-${\mathscr C}$ completion of the fundamental group of $S$. The \emph{${\mathscr C}$-congruence completion $\checkΓ(S)^{\mathscr C}$ of $Γ(S)$} is the profinite completion induced by the embedding $Γ(S)\hookrightarrow{\operatorname{Out}}(\hatπ_1(S)^{\mathscr C})$. In this paper, we begin a systematic study of such completions for different ${\mathscr C}$. We show that the combinatorial structure o…

On monoids of injective partial selfmaps almost everywhere the identity

In this paper we study the semigroup $\mathscr{I}^{\infty}_λ$ of injective partial selfmaps almost everywhere the identity of a set of infinite cardinality $λ$. We describe the Green relations on $\mathscr{I}^{\infty}_λ$, all (two-sided) ideals and all congruences of the semigroup $\mathscr{I}^{\infty}_λ$. We prove that every Hausdorff hereditary Baire topology $τ$ on $\mathscr{I}^{\infty}_ω$ such that $(\mathscr{I}^{\infty}_ω,τ)$ is a semitopological semigroup is discrete and describe the closure of the discrete semigroup $\mathscr{I}^{\infty}_λ$ in a topological semigroup. Also we show that …

扩展双环半群的拓扑化

On topologization of the extended bicyclic semigroup

本文研究了扩展双环半群的$T_1$拓扑构造及其离散条件。

On the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ which is generated by the family $\mathscr{F}$ of atomic subsets of $ω$

We study the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$, which is introduced in [O. Gutik and M. Mykhalenych, \emph{On some generalization of the bicyclic monoid}, Visnyk Lviv. Univ. Ser. Mech.-Mat. \textbf{90} (2020), 5--19], in the case when the family $\mathscr{F}$ of subsets of cardinality $\leqslant 1$ in $ω$. We show that $\boldsymbol{B}_ω^{\mathscr{F}}$ is isomorphic to the subsemigroup $\mathscr{B}_ω^{\Rsh}(\boldsymbol{F}_{\min})$ of the Brandt $ω$-extension of the semilattice $\boldsymbol{F}_{\min}$ and describe all shift-continuous feebly compact $T_1$-topologies on the semigroup $\m…

On variants of the extended bicyclic semigroup

In the paper we describe the group $\mathbf{Aut}\left(\mathscr{C}_{\mathbb{Z}}\right)$ of automorphisms of the extended bicyclic semigroup $\mathscr{C}_{\mathbb{Z}}$ and study the variants $\mathscr{C}_{\mathbb{Z}}^{m,n}$ of the extended bicycle semigroup $\mathscr{C}_{\mathbb{Z}}$, where $m,n\in\mathbb{Z}$. In particular, we prove that $\mathbf{Aut}\left(\mathscr{C}_{\mathbb{Z}}\right)$ is isomorphic to the additive group of integers, the extended bicyclic semigroup $\mathscr{C}_{\mathbb{Z}}$ and every its variant are not finitely generated, and describe the subset of idempotents $E(\mathscr{…

On feebly compact semitopological symmetric inverse semigroups of a bounded finite rank

We study feebly compact shift-continuous $T_1$-topologies on the symmetric inverse semigroup $\mathscr{I}_λ^n$ of finite transformations of the rank $\leqslant n$. For any positive integer $n\geqslant2$ and any infinite cardinal $λ$ a Hausdorff countably pracompact non-compact shift-continuous topology on $\mathscr{I}_λ^n$ is constructed. We show that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $λ$ for a $T_1$-topology $τ$ on $\mathscr{I}_λ^n$ the following conditions are equivalent: $(i)$ $τ$ is countably pracompact; $(ii)$ $τ$ is feebly compact; $(iii)$ $τ$ is $d…

On locally compact semitopological $0$-bisimple inverse $ω$-semigroups

We describe the structure of Hausdorff locally compact semitopological $0$-bisimple inverse $ω$-semigroups with compact maximal subgroups. In particular, we show that a Hausdorff locally compact semitopological $0$-bisimple inverse $ω$-semigroup with a compact maximal subgroup is either compact or topologically isomorphic to the topological sum of its $\mathscr{H}$-classes. We describe the structure of Hausdorff locally compact semitopological $0$-bisimple inverse $ω$-semigroups with a monothetic maximal subgroups. In particular we prove the dichotomy for $T_1$ locally compact semitopological …

双循环幺半群的拓扑化研究

On topologization of the bicyclic monoid

本研究针对双循环幺半群的拓扑化问题，构造了三类非离散/紧致拓扑，给出了其$T_1$拓扑为离散拓扑的判定条件，证明满足Baire、拟正则或半正则性质的逆半群$T_1$拓扑必为离散拓扑。

关于有界有限秩半拓扑对称逆半群的弱紧性注记

A note on feebly compact semitopological symmetric inverse semigroups of a bounded finite rank

本文研究有限秩对称逆半群 $\mathscr{I}_λ^n$ 上的移位连续 $T_1$ 拓扑，刻画弱紧性与序列预紧性的等价关系，并证明弱 $ω$ 有界情形必为紧拓扑。

关于由ω的有限有界区间族F_n生成的半群B_ω^{F_n}

On the semigroup $\boldsymbol{B}_ω^{\mathscr{F}_n}$ which is generated by the family $\mathscr{F}_n$ of finite bounded intervals of $ω$

本文研究由ω的有限有界区间族F_n生成的半群B_ω^{F_n}，证明其Green关系D与J重合、同构于秩≤n+1的部分凸序同构半群且只容许Rees同余，并完整刻画了其移位连续拓扑及Hausdorff半拓扑半群中的闭包和H-闭性。

On topological McAlister semigroups

In this paper we consider McAlister semigroups over arbitrary cardinals and investigate their algebraic and topological properties. We show that the group of automorphisms of a McAlister semigroup $\mathcal{M}_λ$ is isomorphic to the direct product $Sym(λ){\times}\mathbb{Z}_2$, where $Sym(λ)$ is the group of permutations of the cardinal $λ$. This fact correlates with the result of Mashevitzky, Schein and Zhitomirski which states that the group of automorphisms of the free inverse semigroup over a cardinal $λ$ is isomorphic to the wreath product of $Sym(λ)$ and $\mathbb{Z}_2$. Each McAlister se…

不可数多个紧生成单群的局部同构类型

Uncountably many local isomorphism types of compactly generated simple groups

本文解决了局部紧群理论中的开放问题，证明非离散紧生成全不连通拓扑单群类中存在不可数多个（$2^{\aleph_0}$个）局部同构类，并概述了证明思路与来源。